
where \(\bar{\xi }_{\bar{{\cal X}}}\)  is the horizontal lift of \(\xi _{\left[\bar{{\cal X}}\right]}\)  at \(\bar{{\cal X}}\) .
According to [1]}, the Riemannian
Hessian of \(\bar{f}\)  at \(\bar{{\cal X}}\)  can be computed via the Riemannian connection (REF ):
\(\operatorname{Hess}\bar{f}\left(\bar{{\cal X}}\right)\left[\bar{\xi }_{\bar{{\cal X}}}\right]= \nabla _{\bar{\xi }_{\bar{{\cal X}}}}\operatorname{grad}\bar{f}.\) 
